Karthikeyan Shanmugam

FemtoCaching: Wireless video content delivery through distributed caching helpers

We suggest a novel approach to handle the ongoing explosive increase in the demand for video content in wireless/mobile devices. We envision femtocell-like base stations, which we call helpers, with weak backhaul links but large storage capacity. These helpers form a wireless distributed caching network that assists the macro base station by handling requests of popular files that have been cached. Due to the short distances between helpers and requesting devices, the transmission of cached files can be done very efficiently. A key question for such a system is the wireless distributed caching problem, i.e., which files should be cached by which helpers. If every mobile device has only access to a exactly one helper, then clearly each helper should cache the same files, namely the most popular ones. However, for the case that each mobile device can access multiple caches, the assignment of files to helpers becomes nontrivial. The theoretical contribution of our paper lies in (i) formalizing the distributed caching problem, (ii) showing that this problem is NP-hard, and (iii) presenting approximation algorithms that lie within a constant factor of the theoretical optimum. On the practical side, we present a detailed simulation of a university campus scenario covered by a single 3GPP LTE R8 cell and several helpers using a simplified 802.11n protocol. We use a real campus trace of video requests and show how distributed caching can increase the number served users by as much as 400 - 500%.

FemtoCaching: Wireless Content Delivery Through Distributed Caching Helpers

Video on-demand streaming from Internet-based servers is becoming one of the most important services offered by wireless networks today. In order to improve the area spectral efficiency of video transmission in cellular systems, small cells heterogeneous architectures (e.g., femtocells, WiFi off-loading) are being proposed, such that video traffic to nomadic users can be handled by short-range links to the nearest small cell access points (referred to as “helpers”). As the helper deployment density increases, the backhaul capacity becomes the system bottleneck. In order to alleviate such bottleneck we propose a system where helpers with low-rate backhaul but high storage capacity cache popular video files. Files not available from helpers are transmitted by the cellular base station. We analyze the optimum way of assigning files to the helpers, in order to minimize the expected downloading time for files. We distinguish between the uncoded case (where only complete files are stored) and the coded case, where segments of Fountain-encoded versions of the video files are stored at helpers. We show that the uncoded optimum file assignment is NP-hard, and develop a greedy strategy that is provably within a factor 2 of the optimum. Further, for a special case we provide an efficient algorithm achieving a provably better approximation ratio of 1-(1-1/d )d, where d is the maximum number of helpers a user can be connected to. We also show that the coded optimum cache assignment problem is convex that can be further reduced to a linear program. We present numerical results comparing the proposed schemes.

Local graph coloring and index coding

We present a novel upper bound for the optimal index coding rate. Our bound uses a graph theoretic quantity called the local chromatic number. We show how a good local coloring can be used to create a good index code. The local coloring is used as an alignment guide to assign index coding vectors from a general position MDS code. We further show that a natural LP relaxation yields an even stronger index code. Our bounds provably outperform the state of the art on index coding but at most by a constant factor.

Wireless video content delivery through coded distributed caching

We suggest a novel approach to handle the ongoing explosive increase in the demand for video content in mobile devices. We envision femtocell-like base stations, which we call helpers, with weak backhaul links but large storage capabilities. These helpers form a wireless distributed caching network that assists the macro base station by handling requests of popular files that have been cached. We formalize the wireless distributed caching optimization problem for the case that files are encoded using fountain/MDS codes. We express the problem as a convex optimization. By adding additional variables we reduce it to a linear program. On the practical side, we present a detailed simulation of a university campus scenario covered by a single 3GPP LTE R8 cell and several helper nodes using a simplified 802.11n protocol. We use a real campus trace of video requests and show how distributed caching can increase the number of served users by as much as 600-700%.

Finite length analysis of caching-aided coded multicasting

We study a noiseless broadcast link serving K users whose requests arise from a library of N files. Every user is equipped with a cache of size M files each. It has been shown that by splitting all the files into packets and placing individual packets in a random independent manner across all the caches prior to any transmission, at most N/M file transmissions are required for any set of demands from the library. The achievable delivery scheme involves linearly combining packets of different files following a greedy clique cover solution to the underlying index coding problem. This remarkable multiplicative gain of random placement and coded delivery has been established in the asymptotic regime when the number of packets per file F scales to infinity. The asymptotic coding gain obtained is roughly t = K M/N. In this paper, we initiate the finite-length analysis of random caching schemes when the number of packets F is a function of the system parameters M, N, and K. Specifically, we show that the existing random placement and clique cover delivery schemes that achieve optimality in the asymptotic regime can have at most a multiplicative gain of 2 even if the number of packets is exponential in the asymptotic gain t = K(M/N). Furthermore, for any clique cover-based coded delivery and a large class of random placement schemes that include the existing ones, we show that the number of packets required to get a multiplicative gain of (4/3)g is at least O((g/K)(N/M)g-1). We design a new random placement and an efficient clique cover-based delivery scheme that achieves this lower bound approximately. We also provide tight concentration results that show that the average (over the random placement involved) number of transmissions concentrates very well requiring only a polynomial number of packets in the rest of the system parameters.

Graph Theory versus Minimum Rank for Index Coding

We obtain novel index coding schemes and show that they provably outperform all previously known graph theoretic bounds proposed so far 1. Further, we establish a rather strong negative result: all known graph theoretic bounds are within a logarithmic factor from the chromatic number. This is in striking contrast to minrank since prior work has shown that it can outperform the chromatic number by a polynomial factor in some cases. The conclusion is that all known graph theoretic bounds are not much stronger than the chromatic number.

A Repair Framework for Scalar MDS Codes

Several works have developed vector-linear maximum-distance separable (MDS) storage codes that minimize the total communication cost required to repair a single coded symbol after an erasure, referred to as repair bandwidth (BW). Vector codes allow communicating fewer sub-symbols per node, instead of the entire content. This allows non trivial savings in repair BW. In sharp contrast, classic codes, like Reed-Solomon (RS), used in current storage systems, are deemed to suffer from naive repair, i.e. downloading the entire stored message to repair one failed node. This mainly happens because they are scalar-linear. In this work, we present a simple framework that treats scalar codes as vector-linear. In some cases, this allows significant savings in repair BW. We show that vectorized scalar codes exhibit properties that simplify the design of repair schemes. Our framework can be seen as a finite field analogue of real interference alignment. Using our simplified framework, we design a scheme that we call clique-repair which provably identifies the best linear repair strategy for any scalar 2-parity MDS code, under some conditions on the sub-field chosen for vectorization. We specify optimal repair schemes for specific (5,3)- and (6,4)-Reed-Solomon (RS) codes. Further, we present a repair strategy for the RS code currently deployed in the Facebook Analytics Hadoop cluster that leads to 20% of repair BW savings over naive repair which is the repair scheme currently used for this code.

Beyond Triangles: A Distributed Framework for Estimating 3-profiles of Large Graphs

We study the problem of approximating the 3-profile of a large graph. 3-profiles are generalizations of triangle counts that specify the number of times a small graph appears as an induced subgraph of a large graph. Our algorithm uses the novel concept of 3-profile sparsifiers: sparse graphs that can be used to approximate the full 3-profile counts for a given large graph. Further, we study the problem of estimating local and ego 3-profiles, two graph quantities that characterize the local neighborhood of each vertex of a graph. Our algorithm is distributed and operates as a vertex program over the GraphLab PowerGraph framework. We introduce the concept of edge pivoting which allows us to collect 2-hop information without maintaining an explicit 2-hop neighborhood list at each vertex. This enables the computation of all the local 3-profiles in parallel with minimal communication. We test our implementation in several experiments scaling up to 640 cores on Amazon EC2. We find that our algorithm can estimate the 3-profile of a graph in approximately the same time as triangle counting. For the harder problem of ego 3-profiles, we introduce an algorithm that can estimate profiles of hundreds of thousands of vertices in parallel, in the timescale of minutes.

Bounding Multiple Unicasts through Index Coding and Locally Repairable Codes

We establish a duality result between linear index coding and Locally Repairable Codes (LRCs). Specifically, we show that a natural extension of LRCs we call Generalized Locally Repairable Codes (GLCRs) are exactly dual to linear index codes. In a GLRC, every node is decodable from a specific set of other nodes and these sets induce a recoverability directed graph. We show that the dual linear subspace of a GLRC is a solution to an index coding instance where the side information graph is this GLRC recoverability graph. We show that the GLRC rate is equivalent to the complementary index coding rate, i.e. the number of transmissions saved by coding. Our second result uses this duality to establish a new upper bound for the multiple unicast network coding problem. In multiple unicast network coding, we are given a directed acyclic graph and r sources that want to send independent messages to r corresponding destinations. Our new upper bound is efficiently computable and relies on a strong approximation result for complementary index coding. We believe that our bound could lead to an approximation guarantee for multiple unicast network coding if a plausible connection we state is verified.

Distributed Estimation of Graph 4-Profiles

We present a novel distributed algorithm for counting all four-node induced subgraphs in a big graph. These counts, called the 4-profile, describe a graphs connectivity properties and have found several uses ranging from bioinformatics to spam detection. We also study the more complicated problem of estimating the local 4-profiles centered at each vertex of the graph. The local 4-profile embeds every vertex in an 11-dimensional space that characterizes the local geometry of its neighborhood: vertices that connect different clusters will have different local 4-profiles compared to those that are only part of one dense cluster. Our algorithm is a local, distributed message-passing scheme on the graph and computes all the local 4-profiles in parallel. We rely on two novel theoretical contributions: we show that local 4-profiles can be calculated using compressed two-hop information and also establish novel concentration results that show that graphs can be substantially sparsified and still retain good approximation quality for the global 4-profile. We empirically evaluate our algorithm using a distributed GraphLab implementation that we scaled up to 640 cores. We show that our algorithm can compute global and local 4-profiles of graphs with millions of edges in a few minutes, significantly improving upon the previous state of the art.